The word percent comes from Latin and literally means divide (per) by 100 (cent). Even the % symbol, if you look closely at it, resembles /100 . Thus, percents are just another way of writing fractions or decimal fractions. In the chart below are shown the conversions between fractions, decimals, and percents. A more detailed discussion for converting fractions into decimals and decimals into fractions was provided in Unit 3.
As examples consider the following:
As an application of these methods consider the following problem.
The reduction percentage for automotive paint is the amount (volume) of thinner added divided by the amount (volume) of paint originally present. For a 150% reduction how many gallons of thinner must be added to 7.5 gallons of paint?
The solution is that:
There are three basic percent problems.
All three types of problems can be solved by the same strategy of translating word problems into math and a "little bit" of algebra. The translation is as follows:
| what | means |
(an unknown, often called x)
|
| is | means | = |
| of | means | × |
The "little bit" of algebra is as follows. Suppose you wanted to know
from the fact that
=
× 4 = 28.
The "obvious" answer is 7. How did we arrive at this? The opposite of multiplication is division. So
= 28 ÷ 4 = 7.
This solution can be used as a template to solve any similar problem.
=
× a = b, then
= b ÷ a =
The following six problems illustrate the method.
What is 16% of $140 ? (A missing part problem.)
Solution:
= 16% × $140 = 0.16 × $140 = $22.40
27 is 54% of what value? (A missing total amount.)
Solution: 27 = 54% ×
= 0.54 ×
Þ
= 27 ÷ 0.54 = 50.
What percent of $160 is $200?(A missing percent.)
Solution:
× $160 = $200 Þ
= $200 ÷ $160 = 1.25 = 125%.
What percent of
is
?
(A missing percent.)
Solution:
125% of what value is $423? (A missing total amount.)
Solution: 125% ×
= $423 Þ
= $423 ÷ 1.25 = $338.40.
19 is to 80 like what number is to 300?
Solution: This problem is not strictly a percent problem. However, its solution
is very similar in setup to a percent problem.
Here the phrase 19 "is to" translates into the fraction
,
and the phrase "like" translates as = , so this statement becomes
There are many common percent applications. In most communities consumers pay sales tax on certain purchases. If this tax is 5.5%, then the amount of sales tax is (0.055)(sales price). The checkout price is then the sales price +5.5% of sales price or 105.5% of sales price. For example, suppose we purchase an item listed at $89.99
| The sales price | = | $89.99 | = | $89.99 | = | $89.99 |
| The sales tax | = | (0.055)($89.99) | = | $4.94945 | = | $4.95 |
| Checkout | = | $94.94 |
A faster way to compute the checkout price is to compute 105.5% of $89.99 =
(1.055)($89.99) = $94.93945 which, rounded up to the next penny, is $94.94.
Consider the problem below.
The checkout price of a new TV was $258.48 . If 5.5% sales tax was charged, what was the sales price? The solution is as follows:
Purchase Price = 105.5% × Sales Price = 105.5 × Sales Price
Þ Sales Price = Purchase Price ÷ 1.055
Þ Sales Price = $258.48 ÷ 1.055 = $245.00
Occasionally retailers offer discounts on items. A 10% discount means that 10% of the list price has been removed in setting the sales price. If there had been no discount, the sales price would be 100% of the list price. The discount of 10% means that the new sales price is 100% - 10% = 90% of list price. This illustrates an important principle, when considering a percent change, the "base line" is 100%. Thus, a 12% increase means that the new value is 112% of the old value. Working backwards we get the following formula:
Note: multiplying by 100% converts the fractional change into a percent change. If the percent change is positive, this is called a percent increase, if negative, a percent decrease. Consider the following examples.
A VCR regularly priced at $199.99 is discounted 20%. What is the new selling price?
Solution: New Sales Price = (100% - 20%) × $199.99 = 0.80 × $199.99 = $160.00.
If 5.5% sales tax is included, what is the checkout price of a VCR discounted 20%
from its regular price of $199.99?
Solution: Checkout Price = (105.5%) × $160 = 1.055 × $160 = $168.80.
Here, the final answer was rounded up to the next penny.
New car prices of a particular make went from $13,499 to $14,200. What was the
percent increase?
Solution: Percent Increase = (14,200 - 13,499) ÷ 13,499 × 100% = 5.19%.
When a paint is cooled from 75°F to 10°F the viscosity (as measured with a
viscosity cup) increases 89%. At 10°F the time for the paint to
drain from the cup is 49 seconds. How long did it take the paint to drain at 75°F?
Solution: An 89% increase in viscosity means that the low temperature viscosity is
100% + 89% = 189% of the high temperature viscosity.
Drain Time at 10°F = 189% × Drain Time at 75°F = 1.89 ×
= 49 sec
Þ
= 49 sec ÷ 1.89 = 25.9 sec.
When a measurement is specified on a precision part, such as a diameter of 0.150 in plus or minus 0.001 in , the 0.001 in is called the tolerance. This means that any measured diameter between 0.149 in and 0.151 in is acceptable. Another way of specifying the tolerance is the percent tolerance defined by the following formula:
Again multiplying by 100% changes the fractional tolerance into a percent tolerance. The following examples illustrate percent tolerance.
A resistor is rated at 85 W ±
2 W. What is the percent tolerance?
Solution: Percent Tolerance = 2 W &$247;
85 W × 100% = 2.35%.
An acceptable voltage reading is 15.0 V with a percent tolerance of
3%. Express this tolerance in volts.
Solution: Tolerance = 3% × 15.0 V = 0.03 × 15.0 V = 0.45 V.
So the voltage is 15.0 V ± 0.45 V.
Percent error is similar to percent tolerance. When an actual measurement is made and compared to the "true" or "specified" value, the percent error is defined by the following formula:
Usually if the calculated percent error is negative the minus sign is ignored. This is because we are concerned with "how close" we are to the specified value and not whether we are above or below it. The following illustrates a percent error calculation.
A part is specified as having a diameter of 0.250 in. The manufactured part measures
0.254 in. What is the percent error?
Solution: Percent Error = (0.254 in. - 0.250 in.) ÷ 0.250 in. × 100% = 1.60%.
Some sales personnel earn part or all of their salary on a commission basis. This means that they are paid a percentage of their total sales. For example, consider the following situation.
A sales person earns $250 per week plus a 2.5% commission. If the person desires a
week’s gross pay of $900, how much merchandise must be sold?
Solution: The Amount of Commission = $900 - $250 = $650.
2.5% of Sales = 0.025 × Sales = $650
Þ $650 ÷ 0.025 = $26,000.
When getting a loan from a lender, we must pay back both the amount borrowed (the principal) plus interest for the temporary use of the lender’s money. In a simple interest loan everything is paid in one lump sum. For example, if you borrow $1500 at an interest of 5.0%, then you must pay back 100% of the principal plus 5.0% of the principal = 105% of the principal or (1.05)($1500) = $1575
Most conventional loans are paid back in a sequence of monthly payments. In each payment a portion is used to pay the interest owed on the remaining debt and what is left over is used to reduce the debt, i.e., is paid against principal. A conventional loan is characterized by the following four parameters.
To calculate M knowing P, R, and N use the formula:
To calculate N knowing P, R and M use the formula:
To calculate P knowing M, R, and N use the formula:
In the formula for N log is the logarithm function which is found in the second row, fifth column of the Casio fx-300W and the first row, third column of the TI-30Xa. To illustrate these calculations consider the following three examples.
If you borrow $12,000 at an annual percentage rate of 2.3% to be paid off over 4 years,
what is your monthly payment?
Solution:
P = $12,000, R = 0.023 and N = 4, so we need to use the formula for M. Because the number of
instructions that can be stored on the TI-30Xa is less than that of the Casio fx-300W,
the keystrokes for the TI-30Xa must be entered differently than on the Casio
fx-300W. On the Casio fx-300W calculate M with the following keystrokes:
P
R
12
1
1
1
R
12
N
12
For the values given for P, R, and N this works out to be $261.92,
if rounded up to the next penny.
The corresponding keystrokes on the TI-30Xa are:
1
1
1
R
12
N
12
P
R
12
The
key is in the third row, second column.
If you borrow $18,000 at an annual percentage rate of 3.1%
and make monthly payments of $324.24, how long does it take to pay off the loan?
Solution:
P = $18,000, R = 0.031 and M = $324.24, so we need to use the formula for N.
On the Casio fx-300W calculate N with the following keystrokes:
1
R
P
12
M
12
1
R
12
For the values given for P, R, and M the computed value for N is 4.99949 or 5 years.
The corresponding keystrokes for the TI-30Xa are:
1
R
P
12
M
12
1
R
12
If you can afford monthly payments of $350 over a three year period, what is the
most money you can borrow at an annual percentage rate of 4.2%?
Solution:
M = $350, R = 0.042 and N = 3, so we need to use the formula for P.
On the Casio fx-300W calculate P with the following keystrokes:
12
M
R
1
1
1
R
12
N
12
Plugging in the values given for M, R, and N we calculate that P, if rounded
up to the next penny, is $11,819.12. On a practical level you can afford to
borrow about $11,800. The corresponding keystrokes for the TI-30Xa are:
1
1
1
R
12
N
12
12
M
R
Write each of the following numbers as a percent:
0.37 = __________
0.012 = __________
2.59 = __________
= __________
= __________
Write each percent as a decimal number:
22.5% = __________
211% = __________
Write each percent as a fraction reduced to lowest terms:
125% = __________
= __________
Solve for the following unknowns:
What is 16.5% of 128?
answer = __________
125% of $950 is what amount?
answer = __________
53 is what percent of 120?
answer = __________
12 is to 80 like 45 is to what number?
answer = __________
12 is to 60 like what amount is to 90?
answer = __________
$125 is 36% of what amount?
answer = __________
What percent of
is
?
answer = __________
A license cost $120. If the cost increases 7.5%, what is the new cost of this license?
answer = __________
If sales tax is 5.5%, what would be the check out price of a band saw with a list price of $289? (Round to the nearest penny.)
answer = __________
From a 15.0 lb cylinder 1.6 lb of material is removed during machining. What percent of material was removed?
answer = __________
An electrical resistor is rated at 85 ohms plus or minus 5%. Express this tolerance in ohms.
answer = __________
A piston is to have a diameter of 0.787 in ± 0.003 in. What is the percent tolerance?
answer = __________
Specifications call for a pin to be 1.500 in long. If the finished pin measures 1.504 in., what is the percent error?
answer = __________
If an electric drill usually selling for $89.95 is on sale at a discount of 25%, what is the new list price? (Round to the nearest penny.)
answer = __________
A new car is advertised as selling for $14,220. This price reflects a 9% discount. What was the original (list) price? (Round to the nearest penny.)
answer = __________
An assembly line is shut down for inspection if the fraction of defective products exceeds 0.5%. If the normal day’s production is 10,500 units, at most how many defective units can there be if the line is not to be shut down?
answer = __________
A salesperson is paid $380 per week plus a 2.2% commission. What is the person’s sales total if the gross pay for a given week is $1,395? (Round to the nearest penny.)
answer = __________
What is the monthly payment required to pay off a $12,000 loan in two years at an annual percentage rate of 2.7% ? (Round to the nearest penny.)
answer = __________
What is the largest amount which can be borrowed over three years at 4.5% APR if the largest affordable monthly payment is $279? (Round to the nearest ten dollars.)
answer = __________
How long would it take to pay off $15,000 at 5.2% APR if the monthly payment is $450?
answer = __________
You can afford 15% of your monthly income of $2300 on car payments. If the quoted annual percentage rate of the loan is 2.5% over three years, what is the most you can borrow? (Round to the nearest ten dollars.)
 |
© 2002 by Al Lehnen; HTML-ized by Kevin Mirus (kmirus@madison.tec.wi.us or kjmirus@execpc.com). |
| This document was last modified Wednesday, August 21, 2002, 4:10 PM. |